Pressure Transient Test with Sensitivity Analysis

ABSTRACT

Method for using sensitivity analysis to inform the design and performance of a well test are provided. In one embodiment, a method includes providing a reservoir model of pressure transient behavior and performing a sensitivity analysis to identify an input parameter of the reservoir model that can be estimated from pressure transient test data collected from a well location. This method also includes using the results of the sensitivity analysis to design a pressure transient well test for measuring the identified input parameter. Other methods and systems are also disclosed.

BACKGROUND

Wells are generally drilled into subsurface rocks to access fluids, suchas hydrocarbons, stored in subterranean formations. The subterraneanfluids can be produced from these wells through known techniques.Operators may want to know certain characteristics of subterraneanformations penetrated by wells to facilitate efficient and economicexploration and production. For example, operators may perform apressure transient test of the well and use the resulting pressuretransient date to evaluate formation size, shape, and permeability,among other characteristics. A model can be used to estimate variousreservoir parameters from the measured pressure transient data. Forexample, the Warren and Root dual-porosity model can be used for certainreservoirs.

SUMMARY

Certain aspects of some embodiments disclosed herein are set forthbelow. It should be understood that these aspects are presented merelyto provide the reader with a brief summary of certain forms theinvention might take and that these aspects are not intended to limitthe scope of the invention. Indeed, the invention may encompass avariety of aspects that may not be set forth below.

In one embodiment of the present disclosure, a method includes providinga reservoir model of pressure transient behavior. A sensitivity analysiscan be performed to identify an input parameter of the reservoir modelthat can be estimated from pressure transient test data collected from awell location. The method also includes using results of the sensitivityanalysis to design a pressure transient well test for measuring theidentified input parameter.

In another embodiment, a method includes receiving time-varyingcontributions of multiple uncertain parameters of a reservoir model touncertainty in an output of the reservoir model. The time-varyingcontributions in this embodiment are determined through globalsensitivity analysis. The method also includes devising a well test togain additional information about a parameter of the multiple uncertainparameters based on the received time-varying contributions of themultiple uncertain parameters.

In another embodiment of the present disclosure, a computer isprogrammed to identify input parameters of a reservoir model that can beestimated from pressure transient well data. The computer is alsoprogrammed to determine sensitivity indices of the input parameters viaglobal sensitivity analysis and to generate a ranking of the inputparameters according to their respective contributions to variance inoutput of the reservoir model.

Various refinements of the features noted above may exist in relation tovarious aspects of the present embodiments. Further features may also beincorporated in these various aspects as well. These refinements andadditional features may exist individually or in any combination. Forinstance, various features discussed below in relation to theillustrated embodiments may be incorporated into any of theabove-described aspects of the present disclosure alone or in anycombination. Again, the brief summary presented above is intended justto familiarize the reader with certain aspects and contexts of someembodiments without limitation to the claimed subject matter.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features, aspects, and advantages of certain embodimentswill become better understood when the following detailed description isread with reference to the accompanying drawings in which likecharacters represent like parts throughout the drawings, wherein:

FIG. 1 generally represents a wellbore and fracture sets with boundaryelement nodes that may be selected for Fourier series expansion inaccordance with one embodiment;

FIG. 2 is an example of a fracture distribution for a two-dimensionaldiscretely fractured reservoir that may be simulated in accordance withone embodiment;

FIG. 3 depicts the dimensionless logarithmic derivative of wellborepressure for the discretely fractured reservoir of FIG. 2;

FIG. 4 is an example of a fracture distribution, in which the fracturesdo not intersect the wellbore, in accordance with one embodiment;

FIG. 5 depicts the pressure response of the fracture distribution inFIG. 4;

FIG. 6 is an example of a fracture distribution, in which the fracturesdo intersect the wellbore, in accordance with one embodiment;

FIG. 7 depicts the pressure response of the fracture distribution inFIG. 6;

FIGS. 8 and 9 depict the fitting of models to the pressure response ofFIG. 5;

FIG. 10 depicts the fitting of a model to the pressure response of FIG.7;

FIGS. 11 and 12 depict the evolution of principal component analysiseigenvalues for pressure and pressure derivative in the case ofnon-intersecting fractures;

FIGS. 13 and 14 depict the evolution of the principal component forpressure and pressure derivative in the non-intersecting case;

FIGS. 15 and 16 depict cross-plots of the derivative against theparameter to which it is most sensitive at early and late times;

FIG. 17 depicts histograms of the distributions for five inputparameters of a reservoir model for a non-intersecting case inaccordance with one embodiment;

FIG. 18 depicts the standard deviation for predicted pressure andpressure derivative during a well test for the non-intersecting case inaccordance with one embodiment;

FIGS. 19-22 are global sensitivity analysis bin diagrams for pressureand pressure derivative in the non-intersecting case in accordance withone embodiment;

FIG. 23 depicts histograms of the distributions for five inputparameters of a reservoir model for an intersecting case, in which afracture intersects the wellbore, in accordance with one embodiment;

FIG. 24 depicts the standard deviation for predicted pressure andpressure derivative during a well test for the intersecting case inaccordance with one embodiment;

FIGS. 25 and 26 are global sensitivity analysis bin diagrams forpressure and pressure derivative in the intersecting case in accordancewith one embodiment;

FIG. 27 is a block diagram representative of a workflow using globalsensitivity analysis during design, execution, and interpretation of awell test in accordance with one embodiment;

FIGS. 28-30 are flow charts representative of processes enabled bysensitivity analysis in accordance with certain embodiments; and

FIG. 31 is a block diagram of components of a programmed computer forperforming a sensitivity analysis in accordance with one embodiment.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

It is to be understood that the present disclosure provides manydifferent embodiments, or examples, for implementing different featuresof various embodiments. Specific examples of components and arrangementsare described below for purposes of explanation and to simplify thepresent disclosure. These are, of course, merely examples and are notintended to be limiting.

When introducing elements of various embodiments, the articles “a,”“an,” “the,” and “said” are intended to mean that there are one or moreof the elements. The terms “comprising,” “including,” and “having” areintended to be inclusive and mean that there may be additional elementsother than the listed elements. Moreover, any use of “top,” “bottom,”“above,” “below,” other directional terms, and variations of these termsis made for convenience, but does not mandate any particular orientationof the components.

Embodiments of the present disclosure generally relate to the use of asensitivity analysis for a reservoir model to determine how uncertaintyin the input parameters of the reservoir model impacts uncertainty inthe output of the model. The results of the sensitivity analysis can beused to design (i.e., create or modify) a well test plan to facilitateacquisition of information about uncertain parameters. In someembodiments, such a sensitivity analysis is applied to a model for afractured reservoir, though the present techniques may be broadlyapplicable to other reservoirs, such as layered reservoirs or compositereservoirs. With respect to fractured reservoirs, pressure transientdata gathered in most naturally fractured reservoirs tend not to exhibitthe well-known characteristic behavior, including pressure derivative,of the Warren and Root dual-porosity reservoir model. In reality, thereare a rich variety of flow regimes dependent on the fracturedistribution, spatial intensity and fracture conductivity. Consequently,in one embodiment of the present disclosure a semi-analytical solutionfor pressure transient behavior of fractured reservoirs is used to modelthe pressure response of formations with an arbitrary fracturedistribution, density, and conductivity. The fractured system can bedistributed discretely or continuously (network) with conductivitiesranging from very low to infinite. Using the semi-analytical solutionfor fractured reservoirs, a sensitivity analysis can be performed toidentify which reservoir and geological parameters can be estimated frompressure transient test data collected from single or multiple welllocations.

In some instances, a principal component analysis may be employed toexplore the model parameterization and pre-screening the parameters. Aglobal sensitivity analysis (GSA) methodology can then employed todetermine how the uncertainty of each parameter influences theuncertainty in the output from the reservoir model. Application of alinear model can be avoided through this methodology. The results ofcase studies described below indicate that near-wellbore region fractureconductivities have the largest contribution to the total variance ofthe ensemble of output pressure responses whether the well intersectsthe fracture or not. GSA indicates that this parameter may be estimatedindependently from other geomodeling parameters, unlike interpretationsbased on the dual-porosity pressure transient solutions. GSA is thusshown to be an appropriate technique for well test design.

Most geological formations are fractured to some extent as a result ofstress in the Earth's crust. Depending on the stress regime that aparticular formation has been subjected to, these fractures may exist atmany different length scales within the same formation. Fractures andfaults affect many aspects of reservoir management, from drilling andwell placement to production, stimulation, completion, and enhanced oilrecovery (EOR). As a result, much time and effort is spent describingand characterizing the fractures and modeling fluid flow in naturallyfractured reservoirs.

One purpose of modeling fractures is to create dynamic reservoirsimulation properties with the power to predict the reservoir behaviorover time. The trend has been to model the fracture explicitly withinthe geological model to honor the spatial relationships of the naturalfractures. This uses extensive static fracture characterization fromelectrical and acoustic image logs and/or core data. Image logs are usedto quantify fracture densities and orientations along the wellbore,while trends in fracture distribution within the reservoir may bedetermined from a seismic volume attribute process called ant-trackingor geomechanical analysis. Then, fracture data are used to construct adiscrete fracture network (DFN) model which, in addition to thepetrophysical-based geomodeling properties, may be upscaled to generatea second set of properties of permeability, porosity, and a sigma factordescribing the connectivity between the fractures and matrix fordual-porosity dynamic reservoir simulation. Pressure transient tests maybe performed to provide calibration of the reservoir simulationparameters. Traditionally, the characteristic pressure derivativebehavior of the Warren and Root dual-porosity reservoir model isexpected and interpretations based on this model are used to update theupscaled properties in the near well area.

However, an analysis of the Warren and Root model indicates that thispressure transient behavior will be observed just in naturally fracturedreservoirs where the fracture permeability (k_(f)) is of similarmagnitude to the matrix conductivity (k). Field well test data exhibitmany different flow regimes depending on the fracture distribution,fracture density, and fracture conductivity. Most commonly, anycombination of wellbore-storage dominated unit slope, fracture linear,fractured dominated pseudo-steady state, matrix linear, bilinear, etc.,flow regimes are observed on the pressure derivative curves.

It has been shown that there is a general lack of pressure transientsolutions for arbitrarily placed, conductive fractures. Indeed, thenumber of available analytical solutions for naturally fracturedreservoirs is sufficiently limited that fractured basement reservoirshave often been interpreted as dual-porosity systems. To address thisgap, a mesh-free, semi-analytical pressure transient solver for a singlewell or multiple wells is being introduced. This algorithm provides thepressure response of a single well or multiple wells to be obtained forarbitrarily distributed infinite and/or finite conductivity naturalfractures within the reservoir. The fractures can cross each other andintersect with the wells if the geological model stipulates. This newsolution allows the pressure and pressure derivative response of a welltest to be analyzed for a fracture distribution that is consistent withthe actual DFN modeling, where matrix permeability may be much less thanfracture permeability (conductivity).

Estimating fracture properties from pressure transient analysis usingsuch a solution is challenging. The inverse problem (fitting of theobserved pressure data to the analytical solution) is ill-posed and thenumber of degrees of freedom for the new solution is potentiallyinfinite. In addition, these individual parameters cannot be specifiedin a discrete fracture geological modeling exercise with commercialsoftware. Thus, a sensitivity analysis to identify whichfractured-reservoir geological-modeling parameters can be estimated fromthe pressure transient well data can be performed. This type ofsystematic computer experimentation allows the study of the input-outputrelationships in a complicated reservoir model, which involves manyuncertain input variables interacting with each other, and resulting innonlinear output responses. In some embodiments, multiple sensitivityanalysis techniques could be used, such as one that can identifynon-linear behavior (e.g., principal component analysis) and one thatcan provide a measure of pressure variance sensitivity withoutnecessitating linearity or monotonicity of the underlying physical model(e.g., global sensitivity analysis).

Transient Pressure Solutions for Reservoirs with Arbitrarily PlacedConductive Fractures

Consider pressure diffusion for a slightly-compressible single-phasefluid flow in an infinite, isotropic reservoir, where the fluidcompressibility (not the total compressibility of fractures and matrix)and viscosity are assumed constant and the production rate q(t) isassumed to be known: time dependent or constant. The reservoir containsa network of discrete fractures, where Γ_(i) is the i-th fracture withhalf-length l_(i). The fractures may intersect each other and/or thewellbore or may be unconnected to any other feature, except matrixelements. Pressure diffusion in the matrix is described by the followingstandard diffusivity equation (in 2D) as:

$\begin{matrix}{{{{\frac{k}{\mu}\frac{\partial^{2}}{\partial x^{2}}} + {\frac{k}{\mu}\frac{\partial^{2}}{\partial y^{2}}}} = {\varphi \; c_{t}\frac{\partial }{\partial t}}},} & (1)\end{matrix}$

where

=p₀−p(x, y, t), the reservoir pressure change induced by fluidwithdrawal, p₀ is the initial reservoir pressure, and {x, y} areCartesian coordinates. The pressure change

should also satisfy the following conditions at the wellbore (unless thewell intersects a fracture):

$\begin{matrix}{{{\left( {{r_{w}\cos \; \phi},{r_{w}\sin \; \phi},t} \right)} = {p_{w}(t)}},{\phi \in \left\lbrack {0,{2\pi}} \right\rbrack},} & (2) \\{{{r_{w}\frac{k}{\mu}{\int_{0}^{2\pi}{\frac{\partial }{\partial r}\ \left( {{r_{w}\cos \; \phi},{r_{w}\sin \; \phi},t} \right){\phi}}}} = {- {q(t)}}},} & (3)\end{matrix}$

where {r, φ} denote polar coordinates with origin at the wellbore andp_(w) is the uniform wellbore pressure over the surface of thecylindrical wellbore.

The fluid flow inside a fracture Γ_(i) is described as:

$\begin{matrix}{{\left. \left( {\frac{\partial^{2}}{\partial x_{i}^{2}} + {\frac{1}{F_{C\; i}}\left\lbrack \frac{\partial }{\partial y_{i}} \right\rbrack}} \right) \right|_{\Gamma \; i} = 0},} & (4)\end{matrix}$

together with the boundary condition given by

$\frac{\partial }{\partial x_{i}} = 0$

on the extremities of fracture Γ_(i) (i.e., the no-flow condition),where

$F_{C\; i} = \frac{k_{fi}b_{i}}{k}$

is the conductivity of the i-th fracture, b_(i) is its width oraperture, and k is the reservoir permeability. {x_(i), y_(i)} denote thelocal Cartesian coordinate system with x_(i) tangential and y_(i) normalto the direction of the fracture. Integration over the length of thefracture and the boundary conditions on the extremities gives theboundary condition for the incompressible fluid flow inside the fracturethat can be expressed as:

$\begin{matrix}{{\int_{- l_{i}}^{l_{i}}{\left\lbrack \frac{\partial }{\partial y_{i}} \right\rbrack \ {x_{i}}}} = 0.} & (5)\end{matrix}$

In the general case where fractures intersect each other and/or thewellbore, we denote separate groups of connected fractures by I_(j). Thegroup j=0 (i.e., I₀) correspond to the fracture(s) that intersect thewell. In this more general case, and for infinite conductivityfractures, we can reformulate the conditions stated in Eq. 1 to 5 toreflect the uniform pressure condition within the connected fracturegroup and the condition that total flow rate within a group should beq(t) if the well is present in that group. Additionally, we can simplifythe above equations further by introducing the following dimensionlessvariables:

$\begin{matrix}{{t_{D} = \frac{k\; t}{\mu \; c_{t}\varphi \; l^{2}}},{p_{D} = {\frac{2\pi \; {kh}}{\mu \; q_{0}}P}},{x_{D} = \frac{x}{l}},{y_{D} = \frac{y}{l}},{l_{Di} = \frac{l_{i}}{l}},{r_{w_{D}} = \frac{r_{w}}{l}},{r_{D} = \frac{r}{l}},} & (6)\end{matrix}$

where l denotes a reference length (half-length of an average fracture,for example), h is the formation thickness, and q₀ is the reference flowrate. In terms of these dimensionless variables, and for the generalconnected fractured case, the mathematical model defined by Eq. 1 can berewritten as:

$\begin{matrix}{{{\frac{\partial^{2}p_{D}}{\partial x_{D}^{2}} + \frac{\partial^{2}p_{D}}{\partial y^{2_{D}}}} = \frac{\partial p_{D}}{\partial t_{D}}},} & (7)\end{matrix}$

and the conditions at the wellbore and wellbore-intersecting fracturesbecome:

$\begin{matrix}{\mspace{79mu} {{p_{D} = {{p_{D\; w}\left( t_{D} \right)}\mspace{14mu} {on}\mspace{14mu} \Gamma_{i}}},{i \in I_{0}},}} & (8) \\{{{\frac{r_{w_{D}}}{2\pi}{\int_{0}^{2\pi}{\frac{\partial p_{D}}{\partial r_{D}}\left( {{r_{w_{D}}\cos \; \phi},{r_{w_{D}}\sin \; \phi},t_{D}} \right)\ {\phi}}}} + {\frac{1}{2\pi}{\sum\limits_{i \in I_{0}}^{\;}\; {\int_{- 1}^{1}{\left\lbrack \frac{\partial p_{D}}{\partial y_{{D\; i}\;}} \right\rbrack \ {x_{D\; i}}}}}}} = {- {{q_{D}(t)}.}}} & (9)\end{matrix}$

Away from the well, pressure at the connected fractures is given by:

$\begin{matrix}{{\left. \left( {\frac{\partial^{2}p_{D}}{\partial x_{D\; i}^{2}} + {\frac{l_{Di}}{F_{{CD}\; i}}\left( {\left\lbrack \frac{\partial p_{D}}{\partial y_{D\; i}} \right\rbrack - {2\pi {\sum\limits_{k = 1}^{N_{i}}\; {q_{iik}{\delta \left( {x_{D\; i} - x_{{Dii}_{k}}} \right)}}}}} \right)}} \right) \right|^{\Gamma \; i} = 0},} & (10)\end{matrix}$

where q_(iik) corresponds to the flux passing through the i-th fracturefrom the i_(k)-th intersection point and the dimensionless fractureconductivity is defined as

$F_{{CD}\; i} = {\frac{k_{fi}a_{i}}{kl}.}$

Here, we note that pressure continuity should be met at the connectionpoints, i.e. p_(D)(l_(Di)x_(Dii) _(k) )|_(Γi)=p_(D)(l_(Di)x_(Dik) _(k))|_(Γi). Pressure at the unconnected fractures is given by:

$\begin{matrix}{{{\frac{1}{2\pi}{\sum\limits_{i \in l_{j}}^{\;}\; {\int_{- 1}^{1}{\left\lbrack \frac{\partial P_{D}}{\partial y_{D\; i}} \right\rbrack \ {x_{D\; i}}}}}} = 0},{j > 0.}} & (11)\end{matrix}$

Finally, the system definition is completed with the initial conditionat t=0, that is:

p _(D)(x _(D) ,y _(D),0)=0.  (12)

To develop a solution for the above system of equations, the Laplacetransform may be applied. In the Laplace domain, the dimensionlesspressure change induced at the wellbore is expanded using Fourier serieswith modified Bessel functions as the basis functions. The pressurechange at the i-th fracture may also be represented in a similar manner.The number of terms included in the Fourier series expansion can berestricted for computational efficiency. We found N≈10 to 30 elements inthe series sufficient for accuracy and stability of the pressuresolution. The Fourier series expansion is performed for selected nodeson the wellbore and on each fracture. A representation 10 of this schemeis shown in FIG. 1 by way of example. This representation 10 depicts awellbore 12 and fracture sets 14 (here including a set I₀ intersectingthe wellbore 12 and sets I₁ and I₂ apart from the wellbore 12), as wellas boundary element nodes 16 and 18 (for the wellbore 12 and afracture). Although the linear algebra problem to be solved is dense,the number of unknowns is typically much smaller than the numberencountered in a grid-based numerical simulation and so the solution ismore readily achieved. The solution may be found in the time domain. Thesolution technique may also be used to derive transient solutions tomultiple sealing and leaky faults, composite systems with boundaries,constant pressure, no flow and their combinations, etc.

An example of the type of fracture distribution for a two-dimensionaldiscretely fractured reservoir that may be simulated is shown in FIG. 2,with the determined dimensionless pressure and pressure derivativeresponse for the fractured reservoir depicted in FIG. 3. The fracturedistribution depicted in FIG. 2 has a well located at the origin, andFIG. 3 includes various pressure responses 24 and pressure derivativeresponses 26 based on variation of dimensionless fracture conductivity(F_(CD)). The dimensionless fracture conductivity in this case was thesame for each fracture and varied from 10⁻⁴ to 10². We note that thereis little discernible difference between infinitely acting radial flowin a homogeneous reservoir and the fractured reservoir case when F_(CD)is less than 10⁻².

Naturally Fractured Reservoir Geological Modeling Parameters

In the previous section, we outlined a technique for obtaining pressuretransient solutions for arbitrarily placed finite or infinitelyconductive fractures in a homogeneous reservoir. In this section, wedefine the parameters that will be investigated during the sensitivitystudy for the pressure transient behaviour of naturally fracturedreservoirs. The parameters selected for investigation should beconsistent with the physically measurable parameters used in discretefracture modeling processes as these provide a geologically soundrepresentation of the observed fractures. Therefore, we review how theseparameters are derived and characterized.

Natural fracture data are generally collected from openhole logs,particularly wellbore images, and cores. Image logs are most appropriatefor quantifying fracture densities and orientations as thesehigh-resolution resistivity or sonic log images are able to detect bedsand fractures. Fracture analysis using commercial software delivers adescription of tangible fracture attributes such as location, aperture,and orientation. Normally, a combination of automatic and manual traceextraction is performed. An initial quality control (QC) is used to sortfractures that appear open and have large apertures from fractures oflower quality, such as those with smaller aperture or that have healed.The selected fractures are loaded into a geological modeling package forfurther analysis and grouping. Based on how the fractures appear indepth (zones and layers), within certain facies/lithologies and withrespect to orientations observed on a Stereonet plot, fracture sets canbe determined and fracture statistics may be obtained for each set. Foreach fracture set, the following properties are derived from staticdata, ranging from the most certain to the least certain:

-   -   1. Orientation of fractures, including dip, dip azimuth and dip        azimuth dispersion. This is the most well defined property        normally with statistically sufficient data to characterize at        the wellbore control points.    -   2. Fracture spacing. The average spacing of fractures tends to        be consistent, depending on rock type and bed thickness. In        general fracture spacing increases with bed thickness and        behaves differently depending on lithology (spacing in        limestones is considerably greater than in wackes with shale        interbeds). The fracture spacing is converted to modeling        parameter fracture density or intensity. The intensity log is        subjective and will change according to the window length        selected by the interpreter.    -   3. Fracture aperture. The aperture of the fracture is used to        estimate the conductivity of the fracture. The fracture        aperature may be estimated from the borehole image logs, however        borehole conditions may affect the quality of aperature        measurements. Additionally, the conversion to fracture        conductivity using the cubic law or parallel-plate theory        implies a limited fracture fill or variation in aperture. The        natural fracture conductivity is often poorly constrained.    -   4. Estimate of scale and shape. In an ideal world without        truncation, erosion, etc. fractures are thought of as having a        very elongated elliptical shape, but are often modeled as        rectangular parallelepipeds. Fracture height may be understood        well from structural and tectonic history studies; however,        fracture length is poorly controlled. Generally fracture length        is determined from outcrop analogues or layer thickness        considerations.

For pressure transient tests from a single vertical well, where thesystem can be assumed radially symmetrical in 2D, the absolute azimuthaldirection of the plane of the fracture is irrelevant. However, theconcentration or azimuthal dispersion of fracture directions may have animpact on the pressure response due to the potential for increasedfracture intersections at lower concentration (applying von Mises-Fisherdistribution). The expected fracture spacing, $ES, is the measuredparameter and is converted to fracture intensity. As the fractureazimuth direction is constant, the expected spacing is converted tofracture density by

${P_{32} = \frac{C_{P_{3}}}{S_{f}}},$

where S_(f) is the expected fracture spacing and C_(p) ₃ is a constantdependent upon the distribution of the orientation of the fracturesrelative to the line along which the spacing was measured. As we areassuming a single azimuth direction due to radial symmetry of the welltest problem, C_(p) ₃ may be approximated to 1. There is a proportionalrelationship between the dimensionless fracture conductivity F_(CD) andaperture. Aperture is not used directly to control the spatial placementof the discrete fracture planes during modeling. Therefore, for the sakeof simplicity, F_(CD) is considered as a variable parameter and it isscaled with characteristic length, l, which is equivalent to the meanfracture half-length. Finally, fracture length is considered as aparameter. From the above discussion, and the variables of the pressuretransient solution determined in the previous sections, we select thefollowing variables and distributions to perform the sensitivityanalysis of the pressure transient solution to geomodeling parameters.

TABLE 1 Uncertain geomodeling parameters and their distributions.Parameter Parameter Distri- Standard ID Description bution Meandeviation Min Max $KAPPA Fracture Truncated 100 125 10 1000 concen- log-tration normal $ES Expected Truncated 100 25 20 150 spacing, m log-normal $FCD Dimension- Log- 10 40 less fracture normal conductivity$LENGTH Fracture Truncated 100 25 50 200 length, m log- normal

The following reservoir model is defined in addition to the fracturemodeling parameters. A production well is located at (0,0). Thefractures extend within a radially symmetrical domain, with a 500-mradius from the well. The discrete fracture pressure transient solutionoutlined previously generates the results in non-dimensional formaccording to the definitions given in Eq. 6. However, to allow us toexpress result in terms of real time units and pressure with 1-to-1scaling for simplicity, we define the following input parameters:

TABLE 2 Formation and fluid properties. k, md 100 h, m 20 c_(t), bar⁻¹1.421 × 10⁻⁴ μ, cp 1 r_(w), m 0.108 Skin, non-dimensional 0 p_(o), bar500 p_(wfo), bar 500 q, Sm³/day 100 B_(o), Rm³/Sm³ 1.07 φ,non-dimensional 0.1where we assume that the total compressibility is the same for thefracture system and the matrix.

The discrete fracture modeling is performed with a commercial geologicalpackage. Two independent sets of 500 fractures are defined andsimulated. Samples of the resulting fracture distributions and estimateddimensionless pressure and pressure derivative responses are shown inFIGS. 4-7. Two distinct cases are considered: 1) the fractures do notintersect the wellbore (Case 1); and 2) the fractures intersect thewellbore (Case 2). In Case 1, the fractures do not intersect thewellbore. The fracture distribution for Case 1 is depicted in FIG. 4,with the well location positioned at the origin, and pressure responsefor this fracture distribution is depicted in FIG. 5. In Case 2, topreserve the overall fracture statistics within the domain of theexperiment, the fractures of Case 1 are manually translated until theclosest fracture cuts the wellbore at a uniformly uncertain distancealong the wellbore. This parameter will be termed $WELLPOSITION and is avalue from 0 to 1. The resulting fracture distribution and pressuretransient response for Case 2 following this process are shown in FIGS.6 and 7. To avoid numerical stability issues, for any realization wherethe fracture cuts the wellbore less than 2r_(wD) from the tip of thefracture, $WELLPOSITION is set to either 0 or 1.

Set A.284 used for Case 1 and Case 2 described above and represented inFIGS. 4-7 is typical of the fracture distributions prepared for thestudy. In this set member, the fractures are 103.4 m long, F_(CD)=6.8,the fractures are distributed around the wellbore with a density of 0.02(i.e. P₃₂=0.02, an expected spacing of 49.5) and the concentrationK=45.4. We perform a conventional well test interpretation on the SetA.284 Case 1 and Case 2 pressure response.

For Case 1, the pressure derivative response would traditionally beconsidered indicative of a dual-porosity type response, or possiblytriple porosity due to the two valleys. We fit a dual-porosity pseudosteady state model to the data. The two best fit models for Warren andRoot model parameters are shown in FIGS. 8 and 9, with theinterpretation model indicated by the solid lines. In the two possiblematches given, the storativity ratio, ω, equals 0.45 to 0.48 dependingon the valley picked. (More specifically, in FIG. 8, k=112 md, ω=0.45,and the interporosity flow parameter, λ, equals 2×10⁻⁶; in FIG. 9, k=112md, ω=0.48, and λ=1×10⁻⁷.) This suggests that 45 to 48% of the fluid inthe modeled systems is stored in the fracture system. In reality, forSetA.284, 0.0002% of fluid should be in the fracture system (assuming afracture aperture of 0.1 mm or 100 microns). The discrepancy here iscaused by fact that at very early time, infinitely acting radial flow isobserved in the matrix rather than the fracture network.

The Warren and Root-type model is not appropriate to analyze thisfractured reservoir response. The interporosity flow parameter, λ,varies from 1×10⁻⁷ to 2×10⁻⁶. This parameter describes the ability ofthe matrix to flow into the fissures and is also a function of thematrix block permeability and size, i.e., equal to

${\alpha \; r_{w}^{2}\frac{k}{k_{f}}},$

where α is related to the geometry of the fissure network and thecharacteristic size of the matrix block, r_(m). When the matrix blocksare cubes,

$\alpha = \frac{15}{r_{m}^{2}}$

and when the matrix blocks are slabs of thickness

${2r_{m}},{\alpha = {\frac{3}{r_{m}^{2}}.}}$

Low λ describes a very tight matrix or low fracture density, i.e., r_(m)is large. Using the parameters of realization SetA.284 and the slabmodel, we expect λ≈1×10 ⁻¹¹ due to the spacing of the fractures (weacknowledge that λ is often in the range 10⁻⁴ to 10⁻¹⁰ and the lowvalues we observe are related to the discrete fracture problemconsidered). Another aspect of the conventional analysis with a Warrenand Root-type model is that we have no independent measure of fractureconductivity or measure of sensitivity of the result with respect touncertainty in fracture conductivity versus fracture position/fracturedistribution that affect a.

For Case 2, the wellbore intersects a fracture at a point 48% along thefracture, i.e. $WELLPOSITION=0.48. In this case we should be compliantwith the Warren and Root condition that the flow into the wellbore isthrough the fissure network. However, at early time, due to the 2Dgeometry of the problem, we observe bilinear followed by an almostlinear flow regime. This transition is more readily observed in FIG. 7.As generally depicted in FIG. 10, we fit a finite conductivity hydraulicfracture model in a dual-porosity system to estimate the properties suchas the fracture conductivity, F_(C), fracture half-length, x_(f), and ωand λ. Using a finite-conductivity fracture model, an acceptable matchwas obtained with x_(f)=109 m and F_(c)=40500 md.m (and k=105 md,ω=0.40, and λ=1×10⁻⁷). The input parameters of SetA.284 would giveF_(C)=34000 md.m and x_(f)=51.7 m. Here, the conventional analysisoverestimates the fracture length and conductivity due to the proximityof the natural fractures close to, but not intersecting, the wellbore.The dual-porosity parameters are similar to the Case 1 interpretation.Neither conventional analysis provides any quantitative indication ofthe real conductivity of the discrete natural fractures in the reservoiror their distribution.

With the model and sensitivity study parameters defined, and an examplegiven to highlight the difficulty of determining geological modelingparameters including fracture conductivity by conventional analysismethods, in the following section we examine the sensitivity of thewellbore pressure response to the geological modeling fractureparameters in more detail.

Principal Component Analysis

Principal Component Analysis (PCA) fits a multivariate normaldistribution to the input parameters and the pressure response. Theprincipal component is the single linear relationship between the inputparameters and the pressure response that has the highest variance, andtherefore typically represents the combination of input parameters towhich the pressure response is most sensitive.

Before we perform any PCA sensitivity analysis we should decide on whatthe input parameters actually are. In total there are five inputparameters to the fracture generation procedure—the four physicalparameters defined in Table 1, along with a seed for the pseudo-randomnumber generator used in the stochastic discrete fracture modeling. Theinput parameters to our numerical simulator are the location,orientation, length, and conductivity of the fractures. Since the numberof fractures is large, the number of input parameters will also belarge. This discrepancy between the numbers of parameters is aconsequence of the stochastic nature of the fracture modeling procedure.While we can investigate the sensitivity of the response to parametersin Table 1, there may be other properties of the fracture network, notspecified by the fracture modeling procedure, that are equally relevant.

In the model setup where the wellbore does not intersect a majorfracture(s), one property that we expect might have a high impact is thedistance from the wellbore to the closest fracture, which we will denoteas $MINDISTANCE. More generally, the pressure response might besensitive to other nearby fractures that are close either to thewellbore itself or that form a connected path to the wellbore via otherfractures. To represent this we define $MINDISTANCE2 as the minimum of:the difference between the distances of the closest and second closestfractures to the wellbore; the shortest fracture—fracture distance fromthe closest fracture. In either case it represents the additionaldistance to reach another fracture. In principle this definition couldbe extended to investigate the sensitivity of the pressure response tospecific fractures further and further away from the wellbore.

We applied PCA to the pressure responses from 500 realizations of thefracture model whose parameters were selected by Monte-Carlo samplingfrom the distributions given in Table 1. The parameters used in PCA werethe parameters from Table 1, $MINDISTANCE and $MINDISTANCE2 along witheither the pressure or the derivative at a particular time. Since manyof the parameters had log-normal or approximately log-normaldistributions, we used the logarithm of the parameters in PCA. Theparameters were also scaled with their standard deviation before PCA wasapplied. The evolution of PCA eigenvalues for the non-intersecting caseis generally depicted in FIG. 11 (for pressure) and FIG. 12 (forpressure derivative). In these figures, the dashed lines representeigenvalues for the correlation of the input parameters. The evolutionof the principal component for the non-intersecting case is also shownin FIG. 13 (for pressure) and FIG. 14 (for pressure derivative).

As can be seen from FIGS. 11 and 12, the pressure and pressurederivative are sensitive to the input parameters for times greater thanabout 10⁻² hours. FIGS. 13 and 14 show which input parameters have animpact on the pressure and derivative responses. The pressure issensitive to $MINDISTANCE over the depicted time domain, whereas thederivative is just sensitive to $MINDISTANCE for times less than aboutone hour. Both the pressure and the derivative are sensitive to $FCD fortimes greater than one hour. The pressure and derivative do not appearto be particularly sensitive to any of the other parameters, and inparticular are not sensitive to $MINDISTANCE2. This suggests that the$MINDISTANCE in this case is the one parameter determined from thefracture network, but not explicitly used for its generation, thatshould be considered in any further sensitivity analysis.

FIGS. 15 and 16 depict two cross-plots of the derivative against theparameter which it is most sensitive to at early (FIG. 15) and late(FIG. 16) times. The longer solid line in each figure shows theprincipal component, while the shorter solid line in each figurerepresents an additional component. It is clear that the relationshipbetween derivative and the input parameters is not perfectly linear,preventing complete confidence in PCA. Although PCA is useful todetermine parameters which we are unlikely to be sensitive to, and togive a general description of the parameters to which the response issensitive, it cannot here give a reliable quantitative value for thesensitivity. We therefore look at the more robust procedure of globalsensitivity analysis.

Global Sensitivity Analysis

Sensitivity analysis in general looks at quantifying relevance of inputparameters in computing model predictions. However, when the inputparameters are uncertain it is instructive to look at global sensitivityanalysis that quantifies the relation between uncertainties in the inputparameters and uncertainty in the model outcome. Unlike traditionalsensitivity analysis that is based on local partial derivatives, globalsensitivity analysis relies on variance decomposition and explores anentire input parameter space. This can be particularly useful foranalysis of nonlinear and non-monotonic phenomena such as well testinterpretation in DFN reservoirs, where correlation-based analysis isnot applicable.

GSA allows one to quantify individual contributions of the uncertaininput parameters to the total variance of the model prediction viasensitivity indices. We let the uncertainty in the prediction of themodel Y be characterized by its variance V(Y). We will estimate thecontributions to V(Y) due to the uncertainties in the input parameters{X_(i)} that are in turn characterized by their respective variancesV(X_(i)).

For independent input parameters {X_(i)}, the Sobol' decomposition canbe used to represent the variance of the model Y, i.e., V(Y), asfollows:

v(Y)=Σ_(i=1) ^(N) V1+Σ_(1≦i<j≦N) V _(ij) + . . . +V _(12 . . . N)  (13)

where V_(i)=V(E[Y|X_(i)]) are the first-order contributions to the totalvariance V(Y) when X_(i) is fixed (V(X_(i))=0). Since we do not know thetrue value of X_(i) a priori, we should estimate the expected value of Ywhen X_(i) is fixed anywhere within its possible range, while the restof the input parameters X_(˜i) are varied according to their respectiveprobability distributions. Thus,

S1_(i) =V(E[Y|X _(i)])/V(Y)  (14)

is an estimate of relative reduction in total variance of Y if thevariance in X_(i) is reduced to zero (so called Sobol' index). Inpractical terms, S1_(i) quantifies the expected reduction in variance ofmodel prediction Y if the true value of input parameter X_(i) becomesavailable (e.g. through additional measurements). Similarly,V_(ij)=V(E[Y|X_(i), X_(j)])−V_(i)−V_(j) is the second-order contributionto the total variance V(Y) due to interaction between X_(i) and X_(j).

For additive models Y(X), the sum of the first-order effects S1_(i) isequal to 1. This is not the case for the general case of non-additivemodels, where higher order effects (i.e. interactions between two, threeor more input parameters) can play a notable role. The contribution dueto higher-order effects can be estimated by using the total sensitivityindex ST:

ST _(i)=(V(Y)—V(E[Y|X _(˜i) 6]))/V(Y),  (15)

where V(Y)−V(E[Y|X_(˜i)]) is the total variance contribution from theterms in Sobol' decomposition that include X_(i). In the subsequentsections we will demonstrate the utility of these indices by analyzingthe first-order and total contributions to the variance of modelprediction (transient pressure and pressure derivative) at eachexperiment point of interest and providing a time dependent ranking ofthe individual sensitivities. Of course, ST_(i)≧S1_(i), and thedifference between the two represent the relative contribution from thehigher-order interaction effects that include X_(i). A low value of thetotal sensitivity index indicates negligible contribution to the totalvariance due to uncertainty of a given input parameter. Therefore, STcan be used to reduce the dimensionality of the model by assigning afixed value to the identified “irrelevant” parameters.

Application of GSA for Pressure Transient Tests

Here we consider a case when the wellbore is not intersected by afracture and identify the input parameters that contribute the most tothe uncertainty of the predicted pressure transient and pressurederivative during the well test. In the second case, we perform asimilar analysis when the wellbore is intersected by a fracture fromDFN.

The original design of the study is based on three geometric parameterscontrolling DFN and a conductivity of the fracture ($FCD). The DFNparameters include spacing between the fractures ($ES), concentration ofthe fractures in DFN based on von Mises-Fisher distribution ($KAPPA),and the length of the fracture ($LENGTH). The ranges and types ofdistributions for the model parameters are given in Table 1. In thisstudy, a single value of fracture conductivity $FCD was assigned to theentire DFN, and the pressure transient solution is used for pressure andderivative computations.

Case 1: The Wellbore is not Intersected by the Fracture

To investigate the sensitivity of the pressure response to the relativelocation of the wellbore within the fracture network, we introduced anadditional parameter ($MINDISTANCE) representing the distance betweenthe wellbore and the closest fracture. Once the DFN is generated for agiven triplet of $ES, $KAPPA, and $LENGTH, the corresponding value of$MINDISTANCE can be calculated.

Two independent simulation sets each containing 5-by-500 points weregenerated by Monte Carlo sampling. Five additional sets were generatedby substituting a column from the first set by the corresponding columnfrom the second set one-at-a-time to facilitate subsequent GSAcalculations. Overall, 3500 simulations of well test were performed.FIG. 17 shows histograms reflecting distributions of the values of theparameters used in the study. The x-axis is provided with a log10-scale, and just values from two independent sets are shown.

The estimates obtained for uncertainty in predicted pressure transientand pressure derivative exhibited very different behaviors duringdifferent time periods of the tests. This is represented in FIG. 18,which generally shows the evolution of standard deviation for predictedpressure and pressure derivative during the well test. The standarddeviation for pressure monotonously increased with time, with thefastest rate of increase in the middle of the time log-scale. Thelargest uncertainty in pressure was observed at late times of thesimulated well test. The uncertainty in the pressure derivative wentthrough the maximum in the middle of the time log-scale and subsided tonear-zero values at the end of the well test. A pre-screening usingprincipal component analysis was performed to justify the choice ofinput parameters and, more specifically, confirm the sufficiency of$MINDISTANCE in representing the near-wellbore fracture network.

Global sensitivity indices were calculated to investigate which of thefive geomodeling parameters contribute the most to the observeduncertainty in the predicted pressure and pressure derivative and howthis contribution evolves during the well test. First, we presentresults for the first-order sensitivity indices representing individualcontribution to the variance of the well-test prediction due touncertainty in a given input parameter. It is instructive to look at thevalues of the sensitivity indices in the context of the magnitude of thepredicted uncertainty, as shown in FIGS. 19 and 20. These figures areGSA bin diagrams for pressure (FIG. 19) and pressure derivative (FIG.20) and illustrate individual contributions to the total uncertainty(standard deviation) of the model prediction from the uncertain inputparameters. The size of each shaded bin is proportional to the value ofthe first-order sensitivity index (S1) for a given parameter. Theunfilled space between the standard deviation curve and the closest binrepresents the complexity of the model in terms of the portion of thetotal variance that cannot be explained by just the first-order (linear)effects.

According to FIG. 19, uncertainty in the predicted pressure in the earlystages of the test is mainly dominated by uncertainty in $MINDISTANCE.At the later stages of the well test, fracture conductivity ($FCD) playsan increasing role as well, with fracture length being a distant thirdcontributor to the overall uncertainty of the pressure response.Uncertainty in $FCD and $MINDISTANCE together account for almost 90% ofstandard deviation in predicted pressure at the late times.

As shown in FIG. 20, distance to the closest fracture is responsible forthe major portion of uncertainty in pressure derivative in theearly-to-mid log-times including the peak in the value of the totalstandard deviation. At the late time, most of the uncertainty is comingfrom $FCD and, to a lesser extent, $LENGTH and $KAPPA, with contributionfrom $MINDISTANCE becoming negligible.

Due to the observed gaps between the standard deviation curve and theclosest shaded bin in the GSA bin diagrams of FIGS. 19 and 20, a morecomprehensive analysis can be used to account for the higher-ordercontributions to the total variance from the input parameters via totalsensitivity indices. Similar to the first-order analysis, we present thevalues of calculated ST for both pressure and pressure derivative in thecontext of total uncertainty via GSA bin diagrams shown in FIGS. 21 and22. Note that the size of each shaded bin in FIGS. 21 and 22 isproportional to the value of the total sensitivity index for a givenparameter with the sum of ST_(i) for the terms normalized to one.

While the picture for pressure transient remained largely the same asthe one observed for the first-order analysis, the pressure derivativerevealed a more profound early-time contribution to the totaluncertainty from fracture length, combined with $KAPPA and $ES. Overall,the main conclusion from the GSA study for this case is that fractureconductivity and the distance to the near-wellbore fracture are the twomain contributors to the uncertainty of the predicted pressure transientand are likely to be the two most reliably inferred properties from thewell test performed in DFN reservoir. Note that this conclusion is drawnin the context of the ranges and distributions considered for the inputparameters.

FIGS. 19-22 demonstrate the variation in relative sensitivity of eachparameter. Practically speaking, for designing a well test to gainadditional information about an uncertain parameter, we should plan totest for periods where the relative sensitivity is high and sensitivityto other uncertain parameters is low. When the relative sensitivity ofparameters is unchanged, no additional information about the uncertainparameter will be gained and the test time should not be extended.Consideration should also be given to when peak relative sensitivityoccurs: as the total variance decreases, higher resolution measurementsare called for to discern different output responses.

In our study, we observe from FIG. 22 that the minimum distance to thefracture may be most readily determined at ˜10⁻² hr, although there isstill a relatively large contribution from F_(CD). From 1 to 10 hr, therelative sensitivity of F_(CD) is changing with respect to the otherparameter and is reaching maximum sensitivity: this is the optimum timeto determine this uncertain parameter. After 10 hr, there is limitedvalue in continuing the test for additional information as the relativecontribution of the parameters is similar and overall the total varianceof the pressure derivative is reducing and the variance of the pressuredifference is stabilizing. At very late times (>10³ hr), the pressurederivative variance reduces almost to zero as the pressure transientpasses out of the fractured region.

Case 2: Wellbore is Intersected by the Fracture

In this section, we consider a pressure transient test performed in thewellbore intersected by a fracture of a DFN. The fracture networkrealizations were identical to those used in Case 1 and were translateduntil the closest fracture intersects the wellbore at the random(uniformly distributed) location along the fracture length. The relativedistance along the fracture length at the point of intersection($WELLPOSITION) was introduced in the subsequent GSA study along withoriginal four geomodeling parameters given in Table 1. Histogramsreflecting distributions of the values of input parameters for Case 2are shown in FIG. 23. The x-axis is provided with a log 10-scale foreach of the parameters except $WELLPOSITION.

Overall, 3500 simulations for the Case 2 well test setup were performed.The estimated standard deviations for transient pressure and pressurederivative in Case 2 during a well test are provided in FIG. 24 and showvery similar qualitative behavior compared to predictions obtained forCase 1 (see FIG. 18). The magnitude for the monotonously increasingstandard deviation of pressure was almost three times larger than thatin Case 1. The maximum value in standard deviation of pressurederivative in the middle of the time log-scale was almost twice as muchcompared to Case 1.

Cross-plot analysis for both pressure and pressure derivative revealed astrong dependence of the model predictions on $FCD. In both cases,fracture conductivity appeared to have a clear correlation with thetransient predictions, which is expected for this setup. Thisobservation was confirmed by first-order effect GSA results, which areshown in FIGS. 25 (pressure) and 26 (pressure derivative). With the sizeof the shaded GSA bins in these figures proportional to the value of thefirst-order sensitivity index for a given geomodeling parameter, bothplots indicate that $FCD provides the largest contribution to theuncertainty of transient pressure and pressure derivative. Fracturelength and the relative location of intersection with the wellboreaccount for less than 20% of the standard deviation in transientpressure and pressure derivative. Note that in this case first-ordereffects account for the uncertainty (there are no unfilled gaps in theGSA bin diagrams) and therefore calculation of total-effect sensitivityindices can be avoided. Considering FIGS. 25 and 26 from a well testdesign perspective, we note that F_(CD) may be determined by ˜10 ⁻² hr.After ten hours, little additional knowledge will be gained about thisuncertain parameter by extending the flow period or build-up period ofthe well test.

GSA sensitivity indices allow one to rank input parameters according totheir contribution to the uncertainty (variance) of the modelprediction. In both cases given above, the fracture conductivity of thenear wellbore fractures, $FCD, generally had a dominant role inuncertainty contribution for transient pressure and its derivative. InCase 1, near-wellbore fracture location represented via $MINDISTANCE hadan equally relevant contribution to uncertainty in transient pressureover the time domain, but its relevance for pressure derivative waslargely limited to early-to-mid times.

The practical value of GSA analysis relies heavily on the quality andrelevance of the underlying physical model as well as compactness andcompleteness of the problem parameterization. In our case, theinvestigation of the sensitivity of geological modeling parametersduring well test was performed with a forward model that captures thetrue spatial distribution of the fractures. The pressure transientsolution provided the 7000 accurate solutions that enabled thisanalysis.

Another notable aspect of the GSA result is that it indicates that thefracture conductivity of a connected and/or unconnected fracture naturalfracture network may be estimated independently for a realization of adiscrete fracture model and that this should be the first geologicalmodeling parameter adjusted if the spatial distribution and orientationis consistent with the static reservoir characterization data. Ouranalysis of a simulated pressure response of a fracture modelrealization with a conventional dual-porosity model showed thedifficulty of obtaining usable parameters for conditioning or validatingthe fracture model. Indeed, the single hydraulically fractured wellmodel tends to overestimate the fracture conductivity and length due tothe inability to account for the transition from fracture to matrix andfracture dominated flow regimes.

GSA, combined with an accurate forward model, provides a likelymethodology for pressure transient test planning in complicatedgeological environments: fracture, layered, composite, and anycombination of them. In fact, many well-known carbonate reservoirs areboth layered and fractured in denser layers. Sensitivity analysis in ourtest cases highlighted the magnitude difference in sensitivity tofracture conductivity compared to the other parameters. In view of this,tests should be designed to minimize FCD dependency in order to be ableto determine other poorly constrained geological modeling parameterssuch as average fracture lengths.

From the above discussion, it will be appreciated that GSA may beapplied in various ways during the design, execution, and finalinterpretation of a well test, such as a pressure transient test. Forinstance, during pressure transient test design, GSA can be used todetermine the sensitivity of the expected well-test measurement touncertain geological modeling parameters according to an initial testplan. The design of the test plan could then be varied in light of theresults of GSA. In some embodiments, a desired test duration may bedetermined by identifying where the relative sensitivity to a specificuncertain parameter of interest is high or the maximum test duration maybe determined by identifying where the total sensitivity to theuncertain parameters (individually or cumulatively) falls below theresolution of the proposed pressure transient test gauges. The test plancould also be altered and GSA re-performed to improve the relativesensitivity to a specific uncertain parameter of interest. When severaluncertain parameters are identified with GSA to have high sensitivity,complementary measurements can be planned to reduce the uncertainty inone or several of the uncertain parameters. For instance, one embodimentmay include identifying additional measurements to be taken to reduceuncertainty in one or more identified uncertain parameters. Thesensitivity analysis could then again be performed with reduced rangesof uncertainty for one or more of the identified uncertain parametersbased on the additional measurements. Individual contributions of one ormore of the identified uncertain parameters toward variance of pressureor pressure derivative response during a pressure transient well testcould also be compared to the accuracy of the measurement. Further, theinitial test plan may be disregarded if GSA identifies that theuncertain parameter of interest cannot be determined from the proposedtest, thus avoiding “successful failures” in which the pressuretransient test operation is run successfully but the measured datacannot be used for interpretation purposes.

During well test operations and data acquisition, GSA can be performed,with a representative model of the pressure transient test, for theactual conditions experienced (e.g., flow rates, durations, and wellheadpressures) during the pressure transient test. The real time operationof the test could be modified based on the GSA results. For example, inone embodiment the drawdown and build up durations may be shortened orlengthened to increase the relative sensitivity to a specific uncertainparameter of interest. Also, the flow rate and duration of a second flowperiod may be modified if GSA indicates that the relative sensitivity ofa specific uncertain parameter of interest was low during the first flowperiod and build up.

For well test interpretation, GSA may be performed to identify pressuretransient test model parameters of interest prior to the interpretation.As a diagnostic tool, GSA can be used to identify model parameters ofinterest and therefore concentrate the interpretation on these specificparameters. Parameters with total sensitivity values below theresolution of the measurement can be excluded from the interpretation,thus reducing dimensionality of the inverse problem. As a pressuretransient model screening tool, GSA can be used to determine which modelhas a highest total sensitivity for the observed pressure transienttest. Additionally, GSA sensitivity indices can be used to adjust theweighting scheme during interpretation (inversion) including varying theweights as a function of time. A simple example of this can be selectiveinclusion/exclusion of uncertain parameters in the inversion at a givenpoint (e.g., pressure and pressure derivative) during the pressuretransient test.

An example workflow 50 for using GSA during design, execution, and finalinterpretation of a well test is depicted in FIG. 27 in accordance withone embodiment. During the design phase, a geological model (block 52)can be used to identify uncertain parameters (block 54) and a testobjective can be defined (block 56) from the identified parameters. Asshown at block 58, the test duration can be designed and equipmentselected (from available equipment, depicted at block 60) based on GSA.During a real-time data acquisition and monitoring phase, data can beacquired (block 66) and GSA can be used for real-time quality assurance(QA) and updating the test design (block 64). The GSA can be based on amodified geological model (block 62) that can be adjusted in view ofadditional data. On-site interpretation and test reporting can also beprovided (block 68) during the data acquisition. In a finalinterpretation stage, system identification can be performed (block 72)based on the on-site interpretation and a modified geological model(block 70). This stage can also include parameter estimation (block 74)and a final test interpretation (block 76).

From the above description, it will be appreciated that the presentdisclosure introduces various processes relating to the use ofsensitivity analysis in designing and conducting well tests. One exampleof such a process is generally represented by flow chart 80 in FIG. 28.This process includes providing a reservoir model (block 82) andperforming sensitivity analysis for the model (block 84). In at leastsome instances, the reservoir model is a model of pressure transientbehavior of the reservoir during a pressure transient test and thesensitivity analysis is used to identify an input parameter of thereservoir model that can be estimated from pressure transient test data.In one embodiment, the sensitivity analysis includes principal componentanalysis and global sensitivity analysis. The represented processfurther includes designing a well test (block 86) based on the resultsof the sensitivity analysis. In embodiments using a reservoir model ofpressure transient behavior, designing the well test can includedesigning a pressure transient well test for measuring an inputparameter identified through the sensitivity analysis. It is noted thatdesigning the well test can include the initial planning of a well testor varying an existing well test in real time.

Another example of a process introduced above is generally representedby flow chart 90 in FIG. 29. This process includes determininguncertainties of input parameters of a reservoir model using GSA (block92), which can include determining time-varying contributions ofmultiple uncertain parameters of the reservoir model to uncertainty inan output of the reservoir model. Based on these determineduncertainties, a well test can be devised (block 94) to facilitateacquisition of information relevant to at least one of the uncertainparameters. Devising the well test can include selecting a testingperiod based on relative sensitivity of the at least one uncertainparameter compared to sensitivity to other parameters. The well test canbe performed (block 96), such as by conducting a pressure transient welltest.

A further example of a process introduced above is generally representedby flow chart 100 in FIG. 30. This process includes identifying inputparameters of a reservoir model that can be estimated from pressuretransient well data (block 102). Global sensitivity analysis can beperformed to determine sensitivity indices of the input parameters(block 104). Based on this determination, a ranking of the inputparameters according to their respective contributions to variance inoutput of the reservoir model is generated (block 106).

As noted above, the sensitivity analysis described in the presentapplication can be performed with a computer, and it will be appreciatedthat a computer can be programmed to facilitate performance of theabove-described processes. One example of such a computer is generallydepicted in FIG. 31 in accordance with one embodiment. In this example,the computer 110 includes a processor 112 connected via a bus 114 tovolatile memory 116 (e.g., random-access memory) and non-volatile memory118 (e.g., flash memory and a read-only memory (ROM)). Coded applicationinstructions 120 and data 122 are stored in the non-volatile memory 116.For example, the application instructions 120 can be stored in a ROM andthe data 122 can be stored in a flash memory. The instructions 120 andthe data 122 may be also be loaded into the volatile memory 116 (or in alocal memory 124 of the processor) as desired, such as to reduce latencyand increase operating efficiency of the computer 110. The codedapplication instructions 120 can be provided as software that may beexecuted by the processor 112 to enable various functionalitiesdescribed herein. Non-limiting examples of these functionalities includemodeling of fractures and pressure response in a reservoir andperforming principal component analysis and global sensitivity analysis,as described above. In at least some embodiments, the applicationinstructions 120 are encoded in a non-transitory computer readablestorage medium, such as the volatile memory 116, the non-volatile memory118, the local memory 124, or a portable storage device (e.g., a flashdrive or a compact disc).

An interface 126 of the computer 110 enables communication between theprocessor 112 and various input devices 128 and output devices 130. Theinterface 126 can include any suitable device that enables thiscommunication, such as a modem or a serial port. In some embodiments,the input devices 128 include a keyboard and a mouse to facilitate userinteraction and the output devices 130 include displays, printers, andstorage devices that allow output of data received or generated by thecomputer 110. Input devices 128 and output devices 130 may be providedas part of the computer 110 or may be separately provided.

The foregoing outlines features of several embodiments so that thoseskilled in the art may better understand aspects of the presentdisclosure. Those skilled in the art should appreciate that they mayreadily use the present disclosure as a basis for designing or modifyingother processes and structures for carrying out the same purposes orachieving the same advantages of the embodiments introduced herein.Those skilled in the art should also realize that such equivalentconstructions do not depart from the spirit and scope of the presentdisclosure, and that they may make various changes, substitutions andalterations herein without departing from the spirit and scope of thepresent disclosure.

1. A method comprising: providing a reservoir model of pressuretransient behavior; performing a sensitivity analysis to identify aninput parameter of the reservoir model that can be estimated frompressure transient test data collected from a well location; and usingresults of the sensitivity analysis to design a pressure transient welltest for measuring the identified input parameter.
 2. The method ofclaim 1, wherein performing the sensitivity analysis includes performinga global sensitivity analysis to quantify a relationship betweenuncertainties in at least one input parameter of the reservoir model anduncertainty in at least one prediction of the reservoir model.
 3. Themethod of claim 2, comprising using principal component analysis topre-screen for input parameters to which the response of the reservoirmodel of pressure transient behavior is sensitive and then performingthe global sensitivity analysis to quantify the sensitivity of at leastsome of the input parameters, including the identified input parameter.4. The method of claim 1, wherein performing the sensitivity analysisincludes calculating global sensitivity indices for multiple inputparameters of the reservoir model and identifying which of the multipleinput parameters contribute most to expected uncertainty in pressure andpressure derivative predicted by the reservoir model of pressuretransient behavior.
 5. The method of claim 4, wherein performing thesensitivity analysis includes determining how contributions of themultiple input parameters to the expected uncertainty vary over timeduring a pressure transient well test.
 6. The method of claim 1, whereinidentifying the input parameter of the reservoir model includesestimating the individual contribution of the parameter to variance ofpressure or pressure derivative response observed during a pressuretransient well test.
 7. The method of claim 1, wherein providing thereservoir model of pressure transient behavior includes providing areservoir model of pressure transient behavior for a naturally fracturedreservoir.
 8. The method of claim 7, wherein at least some inputparameters of the reservoir model of pressure transient behavior arebased on a discrete fracture network model.
 9. The method of claim 7,wherein the identified input parameter is at least one of fractureconductivity, fracture concentration, expected fracture spacing,fracture length, or a minimum distance from a wellbore to a fracture.10. The method of claim 1, wherein using results of the sensitivityanalysis to design the pressure transient well test for measuring theidentified input parameter includes designing the pressure transientwell test to minimize dependency on an input parameter of the reservoirmodel other than the identified input parameter to facilitatemeasurement of the identified input parameter.
 11. A method comprising:receiving time-varying contributions of multiple uncertain parameters ofa reservoir model to uncertainty in an output of the reservoir model,the time-varying contributions having been determined through globalsensitivity analysis; and devising a well test to gain additionalinformation about a parameter of the multiple uncertain parameters basedon the received time-varying contributions of the multiple uncertainparameters.
 12. The method of claim 11, wherein devising the well testto gain additional information about the parameter of the multipleuncertain parameters based on the received time-varying contributions ofthe multiple uncertain parameters includes selecting a testing periodbased on relative sensitivity of the parameter compared to sensitivityto other parameters of the multiple uncertain parameters.
 13. The methodof claim 11, comprising performing the well test.
 14. The method ofclaim 13, wherein performing the well test includes performing apressure transient well test.
 15. The method of claim 11, whereinreceiving the time-varying contributions of the multiple uncertainparameters includes receiving at least one of a relative and a totalsensitivity of the multiple uncertain parameters at one or moremeasurement points of interest.
 16. The method of claim 11, wherein theparameters are reservoir parameters including at least one of naturalfracture conductivities, reservoir permeability, fault transmissibility,and fracture density.
 17. The method of claim 11, comprising performingglobal sensitivity analysis to determine the time-varying contributionsof the multiple uncertain parameters of the reservoir model touncertainty in the output of the reservoir model.
 18. A computerprogrammed to perform a method that includes: identifying inputparameters of a reservoir model that can be estimated from pressuretransient well data; determining sensitivity indices of the inputparameters via global sensitivity analysis; and generating a ranking ofthe input parameters according to their respective contributions tovariance in output of the reservoir model.
 19. The computer of claim 18,wherein identifying input parameters of the reservoir model includesidentifying parameters of a reservoir model for a fractured reservoir, alayered reservoir, or a composite reservoir.
 20. The computer of claim18, wherein generating the ranking of the input parameters includesgenerating a time-dependent ranking of the input parameters according totheir respective contributions to variance in the output of thereservoir model.